Problem 98 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=\log _{\cos x} \sin x $$

Step-by-Step Solution

Answer

\[\frac{d}{dx}(y) = \frac{\ln \sin x - \tan x \ln \cos x}{(\ln \cos x)^2}\]

Step 1: Apply Logarithm Properties

Use the change of base formula. The change of base formula states that: \[\log _{b}a = \frac{\ln a}{\ln b}\] This changes the function to \[\log_{\cos x}\sin x = \frac{\ln \sin x}{\ln \cos x}\]

Step 2: Apply Chain Rule of Derivation

Differentiating \(\frac{\ln \sin x}{\ln \cos x}\) requires the use of the chain rule and the quotient rule. Let's denote \(u=\ln \sin x\) and \(v=\ln \cos x\). By the quotient rule, \[(u/v)\'= (u\'v - uv\') / v^2\]. Thus, \[(\ln \sin x / \ln \cos x)\'= ((\ln \sin x)\' \ln \cos x - \ln \sin x (\ln \cos x)\' ) / (\ln \cos x)^2\] To differentiate \(\ln \sin x\) and \(\ln \cos x\), we need to apply the chain rule. We have \((\ln \sin x)\' = \frac{1}{\sin x} \cdot \sin x\' = \cos x\) and \((\ln \cos x)\' = - \frac{\sin x}{\cos x} = -\tan x\]. Therefore, the derivative is: \[((\ln \sin x)\' \ln \cos x - \ln \sin x (\ln \cos x)\' ) / (\ln \cos x)^2= (- \cos x \ln \cos x + \tan x \ln \sin x) / (\ln \cos x)^2\]

Step 3: Simplify

To simplify the derivative, get the common denominator on the numerator of the fraction and combine: \[(- \cos x \ln \cos x + \tan x \ln \sin x) / (\ln \cos x)^2= \frac{\ln \sin x - \tan x \ln \cos x}{(\ln \cos x)^2}\].

Key Concepts

Logarithm Properties

Logarithms are an essential concept in mathematics that help us work with exponential functions in a more manageable way. They transform multiplication operations into addition, making calculations simpler. When we talk about logarithm properties, we refer to rules that help manipulate and simplify logarithmic expressions.

Some key properties include:

• Product Rule: \(\log_b (MN) = \log_b M + \log_b N\).
  
• Quotient Rule: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\).
  
• Power Rule: \(\log_b (M^p) = p \cdot \log_b M\).
  
• Change of Base Formula: \(\log_b a = \frac{\ln a}{\ln b}\).
  

In this exercise, the change of base formula is crucial. It allows us to rewrite logarithms in terms of natural logarithms (\(\ln\)), which are often easier to differentiate. This transformation is the first step in solving the exercise.

Chain Rule of Differentiation

The chain rule is a powerful technique in calculus used to differentiate composite functions. A composite function is a function that is made up of two or more simpler functions. The essence of the chain rule is to "chain" the derivatives of these functions together.

For a function \(y = f(g(x))\), the chain rule states that the derivative \(y'\) is given by \(y' = f'(g(x)) \cdot g'(x)\). Essentially, you differentiate the outer function, keeping the inner function unchanged, and then multiply by the derivative of the inner function.
This rule is applied in the differentiation of functions involving logarithms, as seen in this exercise, where you had to differentiate \(\ln(\sin x)\) and \(\ln(\cos x)\). Each required you to apply the chain rule to get to their respective derivatives.

Quotient Rule

The quotient rule is another differentiation technique used when you have a function defined as a fraction of two other functions. If you have a function \(f(x) = \frac{u(x)}{v(x)}\), the derivative \(f'(x)\) is expressed using the formula:

\[(u/v)' = \frac{u'v - uv'}{v^2}\]

This means you take the derivative of the numerator \(u(x)\), multiply it by the denominator \(v(x)\), and subtract the product of the numerator and the derivative of the denominator from it. Finally, the result is divided by the square of the denominator.

In this exercise, differentiating \(\frac{\ln \sin x}{\ln \cos x}\) involved using the quotient rule. Each part is differentiated individually, followed by combination according to the rule. This approach helps systematically find the derivative of more complex, fraction-based functions.

Change of Base Formula

The change of base formula provides a method to convert logarithms of any base to those of another base, commonly used with natural logarithms. It is particularly useful when the base and the argument of a logarithm are not easily workable or don't fit standard bases like 10 or \(e\).

The formula is:
\[\log_b a = \frac{\ln a}{\ln b}\]

By applying this formula, any logarithm can be expressed in terms of natural logarithms, which simplifies differentiation tasks. This technique was the first step in the original exercise where \(\log_{\cos x} \sin x\) was rewritten as \(\frac{\ln \sin x}{\ln \cos x}\). This conversion opens up new methods to differentiate the function using derivative rules applicable to natural logarithms.

The change of base formula is not just a tool for changing bases but also for facilitating further mathematical operations, especially in calculus where natural logarithmic properties are frequently utilized.